Eve Polastri​ works for MI5, organising police protection for high-profile foreign visitors. She’s bored, though she doesn’t entirely know it. The girlfriend of Victor Kedrin, a Russian sex trafficker murdered in Vienna, is put in her charge. Eve arrives at the briefing late, hungover, trying to eat a croissant from a paper bag without making a noise. She blurts out that the killer, who sliced Kedrin’s femoral artery and got away, must have been a woman. ‘Twenty quid it’s a woman,’ she says to her colleague, then has to repeat it to Carolyn Martens, the head of the Russia Section of MI6. ‘No one else could have got that close to him.’ Her irritated boss, Frank, tells her to shut up, but Eve is incapable of shutting up, or of stopping herself, or not overstepping. She has noticed other killings, in other places, that seem unconnected except that in one way or another they appear to be the work of a woman. Frank tells her that CCTV confirms that Kedrin’s killer was a man. Eve doesn’t believe him, and goes to interview the witness (crossing a line, Eve!). The translator can’t get any sense out of the girl – just something about a plank of wood. But Eve secretly records the interview and her Polish husband and his friend burst out laughing when they hear the description of the killer. Ale decha. Small-breasted, like a plank of wood. Definitely not a man.

Viewers of Killing Eve (BBC 3) know this already because they’ve seen Villanelle, formerly known as Oksana Astankova, eating ice cream after the kill, eating it slowly and smiling at a little girl who won’t smile back. On her way out of the café, Villanelle knocks the girl’s ice cream into her lap. Sadism and ice cream, and the soundtrack runs: ‘I’ll never give in to your expectation.’ Eve has noticed Villanelle but no one else has; Martens has noticed Eve, and when she’s fired for failing to protect Kedrin’s girlfriend, murdered by Villanelle (dressed fetchingly as a nurse) in a London hospital, Martens invites her to run a covert investigation into the killer. The title of the show lets you in on the dynamic from the start – the book that it’s based on, compiled from four ebook novellas by Luke Jennings of the Observer, is called Codename Villanelle (John Murray, £7.99). Eve is going to become a target. But it’s nice to know that neither of the main women will die before the end of the first episode. It’s the men who are collateral damage, or might be: the men Villanelle is instructed to kill (by a mysterious group called The Twelve) and the men caught in the crosshairs of Eve’s obsession – her colleagues, her husband. She can’t help it though; she has a thing for female assassins. ‘I’m just a fan!’

Phoebe Waller-Bridge, who also made Fleabag and Crashing, goes in for vaudeville. Her TV shows are bright and outrageous; immoral, grotesque, sexy and unexpected. And sad. But not too sad. There’s a restlessness about them, the relentless unseriousness of the classroom joker. A spy thriller may seem very different territory from her previous shows, domestic comedies which delighted in the everyday oddities and hypocrisies encountered by her young female protagonists. But it isn’t tradecraft and global criminal machinations that interest her so much as what women can get away with. Good girls – and assassins – are meant to blend in. They are not meant to ask for gin and tonics at breakfast meetings, as Eve does, or interrupt important men. Villanelle, played by Jodie Comer, gives Waller-Bridge more scope to show women transgressing, though it’s really only a question of amplification. Villanelle is a killer. She knows how to manipulate people. She speaks multiple languages. Her face speaks several more. She can seduce, but that doesn’t make her predictable. She knows what sort of woman other people – women too – need her to be. Waving a tampon gets her into a bathroom she’s not meant to be in. A Dries Van Noten suit gets her into a Berlin nightclub. She makes an entrance and she slips away. Who’s looking for a pretty assassin in a baseball cap? The novel details the training she underwent to pull off her audacious murders, but in the TV show it’s style over strategy. When Villanelle kills, she likes to watch her victims die. She can’t bear to make it look like suicide: where would be the fun in that? And isn’t it nicer to play with your food before you eat it?

Eve, the wonderful Sandra Oh, is Villanelle’s opposite, and then again she’s not. They see each other for the first time in the bathroom mirrors of the hospital where Eve is protecting Kedrin’s girlfriend. It’s tense for the viewer, but not for either of them. They have no idea who they’re looking at. Villanelle is taken with Eve’s thick wavy hair. ‘Wear it down,’ she says, as Eve fusses with it. Once Eve realises the woman she saw in the bathroom is the assassin, she can’t take her mind off her. When Villanelle discovers Eve is on her trail, the obsession becomes mutual. There’s nothing pious about this Smiley and Karla: Waller-Bridge doesn’t do sentimentality. Everything is undercut. When a scene opens on Eve screaming in bed, she isn’t having nightmares or being attacked; she’s just fallen asleep on her arms and can’t move. Villanelle plays childish tricks on her handler, and on Eve. She’s a psychopath, so that’s no surprise, but the mischief is meta: it’s in the character of the show. One episode begins at night with Villanelle killing a man in a deserted office block. A girl on a bus passing below looks up and sees them, sees the blood on the illuminated window. She quickly dials a number on her phone. ‘Hi mama,’ she says. ‘Do you want me to pick anything up from the shop?’

Part of the comedy comes from the jumbling of registers. Reversing the usual gender roles does a good deal, and Killing Eve makes the most of it. The women murder and scheme and spy while checking their hair, their faces, plucking eyebrows, exchanging lipsticks, borrowing razors. Eve goes to a fancy dinner with toilet paper stuck to a shaving cut on her armpit. There are smoothies. Sometimes it’s just the shock of someone being a bitch. Villanelle watches her elderly landlady struggle down the stairs with the garbage. ‘Allez,’ she coos, patting her legs. ‘Tu vas y arriver!’ There are everyday gags – Eve defending herself with a toilet brush; an awkward eulogy – and less everyday language, which has more to do with Waller-Bridge than her characters: ‘dickswab’, ‘monkey dick’, ‘think-bucket’ (Eve’s boss to Eve: ‘You tiresome think-bucket!’). One character says: ‘I’d nail a cousin to work with that woman.’ Another: ‘It’s disappointing that the mole is the one who looks most like a rodent.’ The women get the good lines, Villanelle most of all, thanks in part to her silly voices and Russian accent. ‘Who are you?’ one victim trembles, close to death. ‘Huuuge queestion!’ she replies.

There’s no flinching from the blood, which is important, but there might be a danger of getting too cute. One night when Eve is at home, alone, Villanelle breaks in to the house. Eve tries to run away, and they wrestle in the bath, but Villanelle isn’t there to kill her; she just wants dinner. They sit down to eat and their strange sexy pas de deux – Eve, soaking wet, doling out leftover shepherd’s pie – is so ridiculous, so neatly pitched between thriller and romance, you don’t know how to react. It’s like being tickled. As Eve’s husband arrives home, Villanelle puts her knife to Eve’s throat, takes her MI6 phone and demands the pin number. This is looking serious. ‘One,’ says Eve, tears streaming down her face, ‘two … um … three… um … four.’ Ba dum.

Killing Eve has the tropes of spy dramas: intertitles (in pink) of European cities; champagne and hotels; Villanelle in hot pants shinning up drainpipes; a reliance on technical wizardry; the constant, underlying suggestion that everyone might be corrupt, that an even greater conspiracy will collapse all our assumptions about good and bad, us and them. A lot of it is absurd, but then the focus isn’t on the mysterious crime syndicate or the double-crossing British secret service. And who can say TV is absurd when the Russians are putting Novichok on doorknobs? Villanelle has a backstory, of course, and as the series ramps up it becomes the grounds for a set-piece finale (the emphasis is shifted from the patriarchal construct of the novel – avenging her murdered father – to something more interesting). Eve has dragged up Villanelle’s old prison files. ‘Why castration?’ she asks. ‘What did he do to you? Tell me.’ ‘OK,’ Villanelle replies, looking at her intently:

‘But can we get one thing clear before going on with this?’

‘Yes.’

‘Is that a sweater … attached to a shirt?’

Can we talk about the clothes? Waller-Bridge cares about them and the clothes in Killing Eve make for some of the best punchlines. Villanelle gets the interesting outfits: Miu Miu bomber jackets and pink tulle dresses, power suits and Doc Marten boots and turquoise pussy bow blouses. They’re models’ clothes, worn with insouciance: the kind of clothes women pay attention to, rather than men. They’re sometimes sexy, but mostly as performance. Sometimes they’re very masculine. Villanelle’s naughtiness, her eccentricity, seems modelled on a modern sort of model, Cara Delevingne perhaps. After all, the glamorous things that are attractive in spy dramas – travel and shopping and restaurants and gadgets – are the same things that are attractive in Vogue, but we only ever get GQ.

Eve doesn’t know how to dress, but not because she doesn’t care. There’s no dutiful feminism here; as in Fleabag, the joke is on skewed female priorities. Villanelle steals Eve’s suitcase and sends it back to London full of expensive clothes, exquisite clothes, a wardrobe she could never afford. What a dream! But is she allowed to wear them? Does she dare? There are moments when it seems like the two women are the only ones who can give each other tender adoration as well as real professional respect. The camera is the third party, staying on their faces in abstracted moments: Villanelle eating an apple; Eve’s hangdog hungover expression (Sandra Oh’s face should win an award). In the course of one hit, Villanelle enters the target’s villa, puts on a dress belonging to his wife and calmly strolls into the middle of a garden party. You expect the guests to turn on her but she seems untouchable. A lovely girl in a garden. It’s unnerving to see a person who’s meant to be undercover walk out into plain sight, though James Bond might get away with it, and it hints at something to do with women’s anxieties about being too visible, being caught out.

It’s not all bad for the men. They’re the nicest characters, even if a few of them have to die. There are sweet young boys on both sides. Sebastian, Villanelle’s adorable neighbour and temporary boyfriend (when she’s trying to be ‘normal’), brings her arnica for her black eye and, thinking she has an abusive relative, promises never to hurt her. He’s the one who should be scared. When an MI6 mole is detected, the man falls to pieces. Carolyn Martens cradles him in her arms while trying to extract the necessary information. ‘I’m sorry you have to see this,’ she says to Eve. Men are useful, but can always be disarmed with sex. Conversely, there’s no real desire for them: that is all directed at other women. The rich psychological territory of women looking at other women is usually given the deadly serious treatment; here it’s deadly but much more fun. Women’s desire comes out in all sorts of ways. They admire themselves; they check out other women on the street; ‘I can’t stop thinking about you,’ Eve tells Villanelle. Villanelle dresses a lover in Eve’s stolen clothes and calls her by Eve’s name:

‘It’s just a biblical fantasy thing.’

‘Do you want me to call you Adam?’

‘No, that’s OK …’

Two-thirds of young straight women admit to sexual fantasies about other women. The real number is probably much higher. Does this tell us something about women or something about the ‘male gaze’? I’m not sure, and Waller-Bridge isn’t slowing down to find out. It’s sensuality rather than sex; we don’t see very much action, though there’s a kind of pornography in watching a man who calls his girlfriend ‘pumpkin’ get machine-gunned to death. Eve’s attraction to Villanelle might ruin her marriage, and though Waller-Bridge doesn’t intend to criticise female desire run amok (it’s not just in the sex but the clothes, the food, the danger), it’s hard not to see there’s a cost. Do you have to be a psychopath to be really free?

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